Student understanding of symmetry and Gauss’s law of electricity
Chandralekha Singh
Department of Physics and Astronomy,
University of Pittsburgh, Pittsburgh, Pennsylvania 15260
arXiv:1602.07376v1 [physics.ed-ph] 24 Feb 2016
Abstract
We investigate the difficulties that students in calculus-based introductory physics courses have
with the concepts of symmetry, electric field, and electric flux which are important for applying Gauss’s law. The determination of the electric field using Gauss’s law requires determining
the symmetry of a particular charge distribution and predicting the direction of the electric field
everywhere if a high symmetry exists. Effective application of Gauss’s law implicitly requires understanding the principle of superposition for electric fields. Helping students learn when Gauss’s
law can be readily applied to determine the strength of the electric field, and then helping them
learn to determine the appropriate shape of Gaussian surfaces if sufficient symmetry exists, can help
develop their reasoning and problem-solving skills. We administered free-response and multiplechoice questions and conducted interviews with individual students using a think-aloud protocol
to elucidate the difficulties that students have with the concepts of symmetry, electric field, and
electric flux. We also developed a multiple-choice test that targets these conceptual issues to obtain quantitative information about their difficulties and administered it to 541 students in the
introductory calculus-based physics courses and to upper-level undergraduates in an electricity
and magnetism course and to graduate students enrolled in a TA seminar course. We find that
undergraduate students have many common difficulties with these concepts.
1
I.
INTRODUCTION
A major goal of most calculus-based introductory physics courses is to help students
develop problem solving and reasoning skills.1–4 Gauss’s law of electricity is an important
topic in the second semester of most calculus-based introductory physics courses. Learning
to reason whether Gauss’s law can be exploited in a particular situation to determine the
electric field, without having to evaluate complicated integrals, can provide an excellent context for helping students develop a good grasp of symmetry considerations. Unfortunately,
students often memorize a collection of formulas for the magnitude of the electric field for
various geometries, without paying attention to symmetry considerations. Students apply
these formulas without being able to differentiate between electric field and electric flux.
They have difficulty identifying situations where Gauss’s law is useful and overgeneralize results obtained for a highly symmetric charge distributions to situations where they are not
applicable. Most textbooks do not sufficiently emphasize symmetry considerations or the
chain of reasoning required to determine if Gauss’s law is useful for calculating the electric
field.
To investigate student understanding of the concepts of symmetry, electric field, and
electric flux, we administered free-response and multiple-choice questions and conducted
interviews with 15 individual students using a think-aloud protocol.5 We then developed a
multiple-choice test with 25 conceptual questions that addresses these issues and administered it to 541 students in the introductory calculus-based physics courses in eight different
classes to obtain a quantitative understanding of the nature of the difficulties. The test
was also administered as a pre- and posttest to undergraduates enrolled in an upper-level
electricity and magnetism (E&M) course and to graduate students enrolled in a TA seminar
course. The tests and interviews explored the extent to which students have become proficient in exploiting symmetry and in making conceptual predictions about the magnitude
and direction of the electric field for a given charge distribution using Coulomb’s or Gauss’s
laws. The test also explores whether students can distinguish between electric field and
electric flux, identify situations in which Gauss’s law can readily be used to calculate the
electric field strength from the information about the electric flux, and the shapes of the
Gaussian surfaces that would be appropriate in those cases.
2
II.
PREVIOUS INVESTIGATIONS RELATED TO ELECTRICITY AND MAG-
NETISM
Investigation of student difficulties related to a particular physics concept is important for
designing instructional strategies to reduce them.6–10 Prior investigations related to electricity and magnetism have included difficulties with general introductory concepts, electrical
circuits, and superposition of the electric field.6–10 Maloney et al.6 developed and administered a 32 question multiple-choice test (the Conceptual Survey of Electricity and Magnetism) that surveys many important concepts covered in the introductory physics courses
and is suitable for both calculus- and algebra-based courses. They found that students have
common difficulties with fundamental concepts related to electricity and magnetism. McDermott et al.7 performed an in-depth investigation of the difficulties students have with
electrical circuits and developed exemplary tutorials and inquiry-based curriculum to significantly reduce these difficulties among introductory physics students and pre- and in-service
teachers.12 Beichner et al.8 have developed conceptual assessments related to electrical circuits. Viennot et al.9 investigated difficulties with the superposition of electric fields by
administering written questions. Belcher et al.10 and Belloni and Christian11 have developed visualization tools to improve student understanding of physics concepts including
those related to electricity and magnetism.
III.
METHOD
Our investigation of student difficulties in discerning symmetry and applying Gauss’s
law was performed using two methods: the design and administration of free-response and
multiple-choice questions to elicit difficulties in a particular context and in-depth audiotaped interviews with individual students using a think-aloud protocol5 while they solved
those problems. A principal advantage of written tests is that they can be administered to
large student populations. Multiple-choice tests are easy and economical to administer and
to grade, have objective scoring, and are amenable to statistical analysis that can be used to
compare student populations or instructional methods. The main drawback is that thought
processes are not revealed completely by the answers alone. However, when combined with
interviews with a subset of individual students, well-designed tests can serve as excellent edu-
3
cational tools. Conceptual multiple-choice tests have already been designed to assess student
understanding of force, energy and momentum, and electricity and magnetism.6,8,13–15 They
show that students’ knowledge of physics is often fragmented and context-dependent and
that students share common difficulties.
IV.
BACKGROUND
Gauss’s law allows us to relate the net electric flux through a closed surface to the net
charge enclosed by the surface.
ΦE = Qenclosed /0 .
(1)
Equation (1) implies that if we know the net electric flux through a closed surface, we can
readily find the net charge inside it. And if we know the net charge inside a closed surface,
we can readily find the net electric flux through it.
In general, Eq. (1) does not mean that we can use Gauss’s law to readily find the magnitude of the electric field |E| at a point. Only in situations where the charge distribution
has very high symmetry can we find |E| from the net electric flux ΦE . Although there are
only three types of symmetry (spherical, cylindrical, and planar) for which Gauss’s law can
readily be exploited to determine the electric field at various points from the information
about the electric flux, students need help in identifying when these symmetries are present.
The net electric flux through a closed surface is given by
I
I
ΦE = E · dA = |E| cos θ|dA|.
(2)
The net electric flux ΦE over a closed (Gaussian) surface can be exploited to determine
the electric field magnitude |E| at an arbitrary point P on the surface only if the following
conditions are satisfied:
1. We can determine the exact direction of E relative to the area vector at every point
on the closed surface by symmetry (only θ = 0, 180◦ or ±90◦ are associated with
sufficiently high symmetry).
2. In some cases, we can divide the closed surface into subsections (for each subsection
the electric flux can be readily calculated for example, the side and two caps of a
cylinder) such that one of the following is true:
4
(a) |E| is the same everywhere on the subsection due to the symmetry of the charge
distribution.
(b) E and the area vector (outward normal to the surface) are perpendicular (θ = 90◦ )
so that there is no electric flux through that subsection.
Thus, to determine if the information about the net electric flux through a closed surface
can be exploited to determine |E| at a point P, we may choose a Gaussian (closed imaginary)
surface such that it contains the point P where we want to determine |E| and |E| cos θ is
known (by symmetry) to have a constant value on each subsection of the surface so that it
R
can be pulled out of the flux integral in Eq. (2). Then |dA| equals the total area of the
subsection of the surface.
Although students can calculate the electric field without regard to symmetry using
Coulomb’s law and the principle of superposition, determining the electric field using Gauss’s
law requires an explicit focus on the symmetry of the charge distribution. The principle of
superposition is also a prerequisite for employing Gauss’s law successfully to determine the
electric field (albeit implicit in the actual application of Gauss’s law) and helps determine
if sufficient charge symmetry exists in a particular situation. In addition to considerations
of symmetry, the area vector and the electric flux are new concepts that are introduced in
the context of Gauss’s law. Students must be able to distinguish the electric flux from the
the electric field, a task that is very difficult. Students need to learn that the electric flux
is a scalar which is positive and negative depending on the relative directions of the electric
field and area vector, while the electric field at a point is a vector. They need to learn that
the electric flux and the electric field have different dimensions, that Gauss’s law applies
only to closed surfaces, and that for any closed surface information about the net enclosed
charge is sufficient to determine the net electric flux through it. On the other hand, the
determination of the electric field at various points due to a charge distribution depends on
the way in which the charges are distributed and can depend on both the charges inside and
outside the closed surface. For example, the electric field at various points on a Gaussian
surface may vary from point to point, even though the net electric flux is zero (and hence
the net enclosed charge is zero).
5
V.
DISCUSSION OF STUDENT DIFFICULTIES
We first administered free-response and multiple-choice questions and interviewed individual students in several calculus-based introductory physics courses about concepts related
to symmetry and Gauss’s law. These investigations provided useful insight into common
difficulties students have with these concepts and also provided guidelines for the development of a 25 question multiple-choice test that was then used as a tool to obtain quantitative
information about the extent to which student have common difficulties with these concepts.
Appendix A provides a summary of the development of the multiple-choice test. The final
version of the test is in Appendix B. This 50 minute test was administered to 541 students
in the calculus-based introductory physics courses in eight different classes at the University
of Pittsburgh. Five additional student volunteers were interviewed (in addition to those
interviewed earlier for a total of 15 individual interviews) to obtain a greater understanding
of their difficulties. Two of these classes were honors courses. The average score on the
test was 49%. Table I shows the percent of students who selected the choices (a)–(e) on
Problems 1–25. The correct responses are italicized. Although some questions have a strong
single distractor, others have several distractor choices that are equally popular. In eight of
the test questions, students had to determine the correctness of three statements. Students
identified the correctness of some of the statements but not others as shown in Table I (we
will discuss these issues in detail later in this section). The reliability coefficient16 α, which
is a measure of the internal consistency of the test, is 0.8, which is considered good by the
standards of test design. The point biserial discrimination16 (PBD) quantifies the ability of a
question to discriminate between students who did well overall and those who did not. This
discrimination index for 16 questions was more than 0.4. Only 4 questions had a PBD value
less than 0.3 (only one question with less than 0.2), which is also good by the standards of
test design.16 The reliability coefficient and discrimination indices show that the conceptual
difficulties with symmetry and Gauss’s law found using the test are meaningful.
Table II shows the concepts that were covered and the questions in the multiple-choice
test that addressed them. The table provides only one of the several ways questions can
be classified. Some of the categories in Table II are subsets of other categories. We have
found these subcategories convenient for classifying student difficulties. The categorization
of concepts in Table II is based on student difficulties and does not necessarily reflect the way
6
experts would categorize those problems. For example, one of the concept categories in which
we placed Problems 13 and 15 is “distinguishing between the electric field and electric flux.”
Although the word “flux” is never explicitly mentioned in these questions, our investigation
shows that students often believed that because there was no charge enclosed inside the
Gaussian surfaces that are exclusively contained in the hollow region, the electric field must
be zero everywhere inside the hollow region. They were often interpreting zero net electric
flux through a closed surface to imply zero electric field at all points of the surface. We
classified Problems 19, 21, and 23 in the same concept category “distinguishing between
the electric field and electric flux.” As shown in Table I, the most common misconception
in Problem 19 was due to confusion between the electric field and the electric flux. In
Problem 21 the most common incorrect response was choice (a) because students believed
that in order to determine the net electric flux through a closed surface, knowledge of
the charge enclosed was not sufficient and charges had to be symmetrically distributed (a
requirement for determining the electric field). The most common incorrect response for
Problem 23 was choice (a), which also arises due to the difficulty in distinguishing between
the electric field and flux. Similarly, we placed Problem 14 in the category “recognizing
symmetry to determine if it is easy to exploit Gauss’s law . . . ,” because many students
incorrectly believed that even in experiment 1, Gauss’s law can be used to infer that the
magnitude of the electric field is the same at points A, B, and C.
A.
The electric charge and electric flux are scalars
The most common difficulty with Problem 1 was mistakenly thinking that the electric
flux and/or electric charge are vectors. In interviews, students justified their response about
why the electric flux is a vector by using the following facts: The expression for flux involves
a scalar product of two vectors. Instead of identifying cos θ as the angle between the electric
field and the area vector, many students concluded that the flux is a vector because it
involves a cos θ. Students pointed to the fact that the electric flux can have both positive
and negative signs. When asked if it would make sense to say that the electric flux points at
30◦ south of west, students often avoided a direct response. Their response implied that for a
physical quantity to be a vector, it was not necessary to be able to specify the exact direction.
Rather, because the electric field lines “going out” of a closed surface contribute positively
7
and those “going in” contribute negatively to the total electric flux through a closed surface,
flux must be a vector. To justify why the electric charge should be a vector, students often
made similar claims that positive charges point outward and negative charges point inward.
It was clear from the responses that students were often referring to the electric field but
calling it “charge.”
B.
The principle of superposition
The performance of many students was closely tied with their understanding of the principle of superposition. For example, Problems 2–4 are related to symmetry and require the
use of the superposition principle to compare the electric field at various points for a given
charge distribution. Many students had difficulty with the principle of superposition and
could not differentiate between the electric field due to individual charges at a point and
the net electric field. Interviews suggest that some believed that only the nearest charge
will contribute to the electric field at a point. Others believed that the magnitude of the
electric field at the desired points in Problem 2 (and 3) should be the same because they
were the same perpendicular distance from the straight line joining the three charges. Some
students provided more detailed reasoning. Instead of viewing it as a problem involving the
addition of three electric field vectors, these students often made guesses by looking at the
distances of points A, B, and C from the three charges and hoping that the electric field
will somehow work out to be the same at the three points. They claimed that in Problem 2,
point A is closer to one charge and farther away from the other two charges than point B
which is equidistant from the two charges and not as far away from the third charge as point
A. Therefore, the electric field at points A and B will be the same if we take into account all
the three charges. Because charge is uniformly distributed on the finite sheet in Problem 3,
this type of confusion was even more prevalent. The most common distractor in Problem 4
implies a similar difficulty. Many students believed that if charges are uniformly distributed
on an insulating equilateral triangle, the magnitude of the electric field will be the same
everywhere on a concentric imaginary triangle.
8
C.
The electric field inside a hollow non-conducting object
Problems 13, 15, and 25 assess student understanding of the electric field inside hollow
non-conducting objects of different shapes due to charges on their surface or charges outside.
Problem 13 was the most difficult question on the test and only 21% of the introductory
students responded correctly. 55% of the students believed that the electric field inside a
non-conducting hollow cube with charge uniformly distributed on its surface will be zero
everywhere. Interviews suggest that some students believed that the hollow region inside is
always shielded from the charges on the surface or charges outside. This notion of shielding
was retained by the students despite being reminded by the interviewer that the object on
which charges are distributed is not conducting. Some students explicitly said that the net
effect of all the charges outside must work out to be zero everywhere inside the hollow region.
One student went on to claim that he has always been amazed at how Gauss’s law can be
used to prove that the electric field in the hollow region inside a closed object is always
zero everywhere, a result that appears to be counterintuitive to him. Some students even
drew spherical or cubic Gaussian surfaces inside a hollow cube to argue that because there
is no charge enclosed, the electric field will be zero everywhere according to Gauss’s law.
Similar to Problem 13, the most common difficulty with Problem 15 was assuming that the
electric field inside the sphere in experiment 1 is also zero everywhere. In interviews and
free-response questions, students used reasoning similar to Problem 13.
For Problem 25, choices (b)–(d) were popular due to the difficulties with the principle of
superposition and the electric field inside a hollow non-conducting sphere. Interviews suggest
that students who believed that the electric field at point A is not zero often thought that
the sphere with the uniform surface charge +Q will produce a larger electric field at that
point because one of its sides is only a distance L away compared to the point charge +Q
which is a distance 2L away. At the end of the interview, in response to the query by some
students, the interviewer discussed with them that for point A, the charge on the sphere can
be thought of as a point charge at the center of the sphere. Students often noted that this fact
is very non-intuitive because of the proximity of one end of the sphere to point A. Students
who claimed that the net electric field at point B is zero often refered to the shielding of
the inside of the sphere from the charges on the sphere and charges outside of the sphere
(similar to Problems 13 and 15) when they were explicitly asked by the interviewer why the
9
point charge near the sphere does not produce an electric field at point B. Even when the
interviewer reminded students that the sphere was non-conducting, they often maintained
that the point charge cannot have any influence inside the sphere. Some students said that
they could not explain exactly why the non-conducting sphere will produce shielding, but
that they remember that the electric field must somehow cancel in the hollow region for all
shapes and charge distributions. Further prodding showed that due to a lack of thorough
understanding, these students were often overgeneralizing or confusing two different facts:
the symmetry argument that shows (using Gauss’s law) that the electric field for a sphere
with a uniform surface charge is zero everywhere inside regardless of whether the sphere is
conducting or insulating, and/or the fact that the electric field inside a conductor is zero in
equilibrium regardless of the shape of the conductor.
D.
The underlying symmetry of a charge distribution
Many students have difficulty realizing that it is the symmetry of the charge distribution
(and not the symmetry of the object on which the charges are embedded) that is important
in determining whether Gauss’s law can be applied to calculate the electric field at a point.
For example, students had to determine when a Gaussian surface would be convenient
for determining the electric field at a point on its surface in Problems 6, 11, and 22. In
Problems 11 and 22 they had to identify the shape of appropriate Gaussian surfaces that
would make it easy to use Gauss’s law to calculate the electric field due to an infinite
uniform sheet of charge and line of charge respectively. In Problem 11 the most common
distractor was (e), and many students believed that all surfaces will work because they are
all symmetric. However, the calculation for the sphere is not easy because the area vector
and the electric field make different angles for different infinitesimal areas on the sphere. In
Problem 6 students were asked to evaluate the validity of three general statements without
being given a specific charge distribution that produced the electric field. Many students
chose (e) and believed that the Gaussian surface must be chosen to take advantage of the
symmetry of the object enclosed, regardless of how the charges are distributed on that
object.
The idea of whether the symmetry of the charge distribution or the symmetry of the
object on which charges are distributed is important for being able to determine the electric
10
field using Gauss’s law is also explored in Problems 10 and 14–16. In Problem 10 Gauss’s
law can readily be used to determine the electric field in cases (i) and (iii) because of the
spherical symmetry of the charge distribution on nonpolarizable objects but not in case (ii).
In Problems 14–16 students were presented with an insulating sphere on which there are six
point charges distributed in a way that the adjacent charges are equidistant. Although the
charges are on a spherical object, the charge distribution does not have spherical symmetry.
Interviews and written responses suggest that many students incorrectly believed that a
spherical symmetry exists in this case for exploiting Gauss’s law to calculate the electric field
readily. The most common difficulty with Problem 14 was assuming that the magnitude of
the electric field in experiment 1 would be the same at the three points shown. In interviews
and free-response questions, some students explained this respnse by claiming that because
the six point charges and the three points are symmetrically situated, the field magnitude
must be the same at the three points; others explained their response by claiming that for a
point outside, the six point charges on the sphere can be thought to be point charges at the
center of the sphere. Problem 16 is one of the most difficult questions on the test. It was
easy for most students to rule out (i), but it was difficult for them to evaluate the validity of
(ii) and (iii) because of the difficulty of recognizing the underlying symmetry of the charge
distribution.
Problems 17 and 22 also probe the extent to which students can discern the underlying
symmetry of the charge distribution. In Problem 17 the last three distractors were popular
and students often believed that we can use Gauss’s law to find the electric field at a point
outside due to a cube or finite cylinder with uniform surface charge. In the interviews
students sometime recalled using Gauss’s law for these surfaces. More prodding showed
that they were either confusing the fact that those surfaces can be used as Gaussian surfaces
for appropriate charge distributions or the fact that for an infinite cylinder (but not a finite
cylinder) it is possible to exploit Gauss’s law to find the electric field. It appears that many
students have not thought carefully about the principle of superposition and its implication
for the electric field due to a charge distribution and were applying memorized knowledge
whose applicability was forgotten. In Problem 22 students had to choose the Gaussian
surfaces that would help them determine the electric field at point P readily due to the
infinite line of charge. All of the alternative choices were selected with an almost uniform
frequency. Students were often unsure about the symmetry concepts relevant for making
11
appropriate decisions and those who chose option (c) were often quite confident that the
magnitude of the electric field due to the infinite line must be the same at every point on
the cube as well.
E.
Electric field and electric flux
Problems 7, 8, and 9 are related to electric flux and the distinction between the electric
field and flux. The most common distractor in Problem 8 was (d). Some students were
quite assertive in their interview and incorrectly claimed that if the magnitude of the electric
flux through one closed surface is smaller than another, the magnitude of the electric field
at points on the first surface must be smaller too. All the distractors were popular for
Problem 9. Interviews suggest that students who chose (c) often believed that although
the net charge enclosed is the same for surfaces A and B, the electric flux through surface
A must be larger because it is closer to the positive charge at the center. Interviews also
suggest that students who selected option (d) often believed that although the net charge
enclosed is the same for surfaces A and B, the electric flux through surface B must be larger
because it has a larger area.
Similar difficulty in differentiating and relating the electric field and electric flux was
manifested in response to Problems 12, 18, 21, 22, and 23. The most common difficulty with
Problem 12 was assuming that both electric flux and electric field can be determined using
Gauss’s law; 30% of the students chose option (c). Interviews suggest that students were
often confident about their choice of (c) because the surface area of the Gaussian sphere was
H
given and they believed that Φ = |E|A is always true (instead of Φ = E · dA). Problem 18
is more abstract than most other questions in which a specific physical situation is explicitly
given. In this question students had to evaluate the validity of three statements and the main
difficulty was not being clear about the relation and distinction between the electric field and
electric flux. The most common difficulty with Problem 19 was assuming that the electric
field is zero at point B on the side surface of the cube, although the problem statement
explicitly mentions that the cube is in a uniform electric field of 20 N/C. In interviews and
free-response questions, some students explicitly claimed that the area vector of the side
surface is perpendicular to the direction of the electric field lines. Therefore, the electric
field must be zero at point B. This kind of confusion between the electric field at a point
12
and the contribution to the electric flux from a certain area was found in other questions as
well.
Response to Problems 21 and 22 suggest that many students are not comfortable with
the statement of Gauss’s law that relates the net flux through a closed surface to the net
charge enclosed. They have difficulty differentiating between the electric flux through a
closed surface and the electric field at a point on the surface. For example, in Problem 21
many students chose (i) (or (i) and (iii)) and claimed in the interviews and free-response
questions that only those surfaces can be used to determine the net electric flux through them
because the other surfaces did not have the correct symmetry. The most common distractor
in Problem 23 was (a), which was chosen by 33% of the students. These students believed
that the net charge enclosed in a region is largest if the number of field lines penetrating
the region is greatest. They did not pay attention to the direction of the electric field lines
which is crucial for determining the net flux through a closed surface and hence the net
charge enclosed using Gauss’s law.
F.
Other Difficulties
Students were sometimes unsure about the distinction between open and closed surfaces
and that Gauss’s law is only applicable to closed surfaces. Problems 5, 21, and 22 at
least partly assess whether students understand this distinction. Some students incorrectly
believed that Gauss’s law applies to any symmetrical surface even if it is not closed. For
example, in response to Problem 21, these students claimed that the electric flux Φ due to
an infinitely long line of charge (with uniform linear charge density λ) is λL/0 even for the
two-dimensional square sheet.
In response to Problem 24 many students selected some of the statements as correct but
only 29% identified that all three statements are correct. Many had difficulty determining
the validity of statement (ii) which can be checked by drawing a Gaussian surface in region A
that includes the center and using Gauss’s law. In the interview a student insisted that there
can be a point charge in region A even if the electric field is zero. When asked to explain, the
student drew a positive point charge with the electric field lines radially outward and said:
“well . . . the field due to a point charge will cancel out because it points in all directions.”
It was clear from the explanation that the student was confused about the electric field line
13
representation.
G.
Performance of Upper-Level Undergraduates
We also administered the 25-item multiple-choice test as a pre- and post-test to students
enrolled in a sophomore-junior level E&M course in which vector calculus was used extensively; 33 students took the pretest and 28 students took the posttest. Tables III and IV
show the percentage of students who made the various choices on various questions on the
pretest and posttest respectively. The average pre- and posttest scores are 44% and 49%
respectively; this difference is not statistically significant. The fact that the upper-level
undergraduates did not perform better than the introductory students (compare Tables I
and IV) suggests that the higher mathematical sophistication of the course with vector calculus did not help students acquire a better conceptual understanding of the superposition
principle, symmetry, and Gauss’s law.
H.
Performance of Graduate Students
To calibrate the test we also administered it (over two consecutive years) to a total of
33 physics graduate students who were enrolled in a seminar course for teaching assistants
(TAs). Most of them were first year graduate students who were simultaneously enrolled
in the first semester of the graduate E&M course. Students were told ahead of time that
they would be taking a test related to electrostatics concepts. They were asked to take the
test seriously, but it did not count for their course grade. The average test score for the
graduate students was approximately 75% with the reliability coefficient α ≈ 0.8.16 The
better performance of graduate students compared to the undergraduates is statistically
significant. The minimum score obtained by a graduate student (an American student) was
28% and the maximum score obtained by two Chinese graduate students was 100%.
Table V shows the percentage of graduate students who selected the various choices
on the test. Many of the conceptual difficulties that the graduate students displayed are
similar to those of the introductory students. Problems 13 and 17 were the only ones on
which graduate students performed less than 50% with misconceptions similar to those of
the undergraduates. The PBD for Problems 13 and 17 were 0.5 and 0.6 respectively, which
14
shows that the graduate students who performed well overall on the test did well on those
questions. Comparison of Tables I and V shows that on an average, the graduate students
outperformed the introductory students on all other questions except Problem 2, but this
difference is not statistically significant. The PBD for this question is 0.5, so the graduate
students who performed well overall on the test did well on this question.
VI.
SUMMARY
We find that the undergraduate students, including those in the upper-level course with
higher mathematical sophistication, have common difficulties related to the superposition
principle, symmetry concepts, and Gauss’s law. The knowledge deficiencies can be broadly
divided into three levels with increasing difficulty: lack of knowledge related to a particular
concept, knowledge that is retrieved from memory but cannot be interpreted correctly, and
knowledge that is retrieved and interpreted at the basic level but cannot be used to draw
inferences in specific situations. Our investigation shows evidence that students’ difficulties
were due to knowledge deficiencies across all three levels. Because many questions required
students to predict the outcomes for specific setups, they necessitated a transition from a
mathematical representation to a concrete case. Therefore, deficiencies at the latter two
levels were frequently observed. Instructional strategies that focus on improving student
understanding of these concepts should take into account these difficulties. The multiplechoice test that we developed can help assess the effectiveness of strategies to improve student
understanding of these concepts.
Acknowledgments
We are very grateful to P. Reilly and all the faculty who reviewed the various components
of the test at various stages and provided invaluable feedback. We are also very thankful
to all the faculty who administered the test. We thank R. Devaty, A. Janis, R. Johnsen,
and J. Levy for a critical reading of the manuscript. This work is supported in part by the
National Science Foundation award DUE-0442087.
15
VII.
TABLES OF RESULTS
Q 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
(a) 55 3 20 0 14 3 5 4 7 20 4 57 55 12 40 4 0 47 2 22 28 37 35 10 32
(b) 35 13 57 30 0 11 6 50 66 0 5 6 10 24 8 10 33 7 3 14 5 15 2 15 33
(c) 5 69 15 5 0 8 81 5 12 19 57 30 8 2 43 18 25 27 34 6 17 13 53 27 22
(d) 1 7 7 15 0 58 6 35 5 50 6 7 6 60 4 42 17 13 56 53 32 17 3 19 12
(e) 4 8 1 50 86 20 2 6 10 11 28 0 21 2 5 26 25 6 5 5 18 18 7 29 1
TABLE I: Percentage of introductory calculus-based physics students (total number of students
541) who selected choices (a)–(e) on Problems (1)–(25) on the test. The correct response for each
question has been italicized. The average score was 49%.
16
Concepts
Problem number
Electric flux
1, 7, 8, 9, 12, 18, 20, 21, 23
Recognizing the symmetry of the charge distribution 2, 3, 4, 6, 10, 11, 12, 13, 14, 15,
16, 17, 22, 24, 25
Symmetry of the object versus symmetry of the charge 6, 10, 14, 15, 16
distribution
Coulomb’s law, superposition and symmetry consider- 2, 3, 4, 14, 16
ations sufficient
Difference between the electric field and the electric 1, 7, 8, 12, 13, 15, 18, 19, (21, 22),
flux
23
Relevance of a closed surface in Gauss’s law
5, 21, 22
Recognizing the symmetry to determine if it is easy 6, 10, 11, 12, 13, 14, 15, 16, 17,
to exploit Gauss’s law or exploiting Gauss’s law to 22, 24, 25
determine the electric field
Appropriate Gaussian surface for determining the 6, 11, 22
electric field for a given charge distribution
Electric field inside hollow non-conducting objects 13, 15, 25
with different charge distribution
TABLE II: Concepts covered and the questions that addressed them in the multiple-choice test.
Q 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
(a) 42 9 15 3 30 0 9 0 9 24 3 43 49 9 49 9 0 58 0 15 15 27 42 12 27
(b) 49 15 61 36 0 3 18 52 67 0 15 18 15 24 3 21 27 3 6 18 13 15 3 27 30
(c) 6 52 9 9 0 3 64 12 15 21 24 33 15 9 36 27 30 9 24 9 15 12 42 30 21
(d) 3 21 15 18 0 82 0 24 6 45 18 3 3 55 3 33 6 18 64 55 36 30 3 12 21
(e) 0 3 0 33 70 12 9 12 3 9 39 3 18 3 9 9 36 12 6 3 21 15 9 18 0
TABLE III: Percentage of students in the upper-level undergraduate E& M course (total number
of students 33) who selected choices (a)–(e) on the pretest (before instruction in the upper-level
course). The correct response for each question is italicized. The average score was 44%.
17
Q 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
(a) 53 11 11 4 32 0 11 11 4 21 0 53 75 0 64 4 0 39 0 7 7 50 36 4 28
(b) 39 29 61 46 0 11 11 32 68 4 4 11 0 32 4 7 7 7 4 21 4 4 4 7 32
(c) 0 57 11 7 0 4 68 7 18 11 68 25 4 4 25 7 43 32 36 14 32 18 53 18 0
(d) 4 0 14 4 0 82 11 36 7 50 0 7 7 64 7 46 11 11 53 53 53 14 0 11 36
(e) 4 4 4 39 68 4 0 14 4 14 28 4 14 0 0 36 39 11 7 4 4 14 7 61 4
TABLE IV: Percentage of students in the upper-level undergraduate E& M course (total number
of students 28) who selected the choices (a)–(e) on the posttest (after instruction in the upper-level
course). The correct response for each question is italicized. The average score was 49%.
Q 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
(a) 73 3 12 0 12 3 3 6 0 12 0 79 55 0 36 6 0 73 3 3 6 73 18 0 55
(b) 27 9 79 18 0 3 3 88 91 0 3 0 0 12 3 6 45 0 0 3 3 3 0 6 9
(c) 0 64 9 0 0 6 91 0 0 6 94 12 9 3 61 9 40 24 0 0 0 12 82 9 12
(d) 0 3 0 9 0 76 0 6 6 70 0 6 0 85 0 27 0 0 94 94 88 12 0 30 24
(e) 0 21 0 73 88 12 3 0 3 12 3 3 36 0 0 52 15 3 3 0 3 0 0 55 0
TABLE V: Percentage of physics graduate students enrolled in a course for teaching assistants
(total number of students 33) who selected the choices (a)–(e) on Problems (1)–(25) on the test.
The correct response for each question has been italicized. The average score was 75%.
18
Appendix A: Summary of the Multiple-choice Test Design
During the design of the multiple-choice test, we paid particular attention to the issues
of reliability and validity.16 Reliability refers to the relative degree of consistency between
testing if the test procedures are repeated for an individual or group. Validity refers to the
appropriateness of the test score interpretation. The test design began with the development
of a test blueprint that provided a comprehensive framework for planning decisions about
the desired test attributes. The degree of specificity in the test plan was useful for creating
questions. We tabulated the scope and extent of the content covered and the level of
cognitive complexity desired. We used previous free-response and multiple-choice questions
administered to students as a guide and identified the desired performance and a description
of conditions/contexts under which the performance was expected to occur.
We classified the cognitive complexity using a simplified version of Bloom’s taxonomy17 :
specification of knowledge, interpretation of knowledge, and application of knowledge in
different situations. The performance targets and table of content and cognitive complexity
were shown to five physics faculty members at the University of Pittsburgh for review.
Modifications were made to the weights assigned to various concepts and to the performance
targets based on faculty feedback. The performance targets were converted to approximately
30 free-response questions. These questions required students to provide reasoning for their
responses. The free-response questions were administered (in groups of 10–20) to students in
the calculus-based courses. Often, some students in a class were given one set of questions
and others were given another set in order to sample student responses on most of the
questions. We also tape-recorded interviews with 5 introductory student volunteers using
a think-aloud protocol.5 30 multiple-choice questions were then designed using the most
frequent incorrect student responses for the free-response questions and interviews as a guide
for making the distractor choices. Choosing the four distractors to conform to the common
difficulties is essential for increasing the discriminating properties of the questions. Five
physics faculty members were asked to review the multiple-choice questions and comment
on their appropriateness and relevance for calculus-based introductory physics and to detect
ambiguity in question wording. A review form was developed to aid the faculty in reviewing
the questions. The faculty also classified each question on a scale from very appropriate to
least appropriate. Further modifications were made based on their recommendations. Then,
19
a multiple-choice test was assembled using 25 questions that closely matched the initial table
delineating the scope of the content and cognitive complexity. The same faculty members
who earlier reviewed the questions were shown the test several times and modifications were
made based on the feedback during the iterations.
The 50 minute multiple-choice test was administered after instruction in Gauss’s law
to students in the calculus-based courses at Pittsburgh. Five student volunteers who had
not taken the test earlier were interviewed individually and asked to respond to the test
questions using the think-aloud protocol. These interviews provided us with an opportunity
to clarify issues in-depth. The reliability index α for the test was approximately 0.8 which
is good by the standards of test-design.16 Item analysis of student responses was performed
to judge whether each question functioned as expected.16 In addition to the calculation
of difficulty and discrimination of questions,16 item analysis included creating a table to
count the number of students selecting each distractor in the upper and lower quartiles.
Item analysis was very useful to determine whether individual questions and distractors
functioned as expected. Based upon the item analysis and interviews, the test questions
were modified further before being administered to the students.
20
Appendix B: The multiple-choice test
Instructions: Select one of the five choices (a)–(e) for each of the 25 Problems. In all questions
all physical objects are insulating (non-conducting); all insulators are nonpolarizable; 0 is the
H
~ cos θ = Qenc /0 ; and the sign convention is that
~ dA|
permittivity of free space; Gauss’s law: |E||
for all closed surfaces, consider outward flux as positive.
Problem 1. Choose all of the following physical variables that are vectors:
(i) Electric field
(ii) Electric flux
(iii) Electric charge
(a) (i) only.
(b) (i) and (ii) only.
(c) (i) and (iii) only.
(d) (ii) and (iii) only.
(e) (i), (ii), and (iii).
Problem 2. Three identical point charges +Q are arranged in a line as shown in Fig. 1.
Points A, B, and C are along a parallel line. You do not know the lengths L and
d. The three charges produce an electric field. Without knowledge of L and d,
what can you infer about the electric field at points A, B, and C?
(a) Both the magnitude and direction of the field are the same at points A, B,
and C.
(b) The magnitude of the field is the same at points A, B, and C but the
directions are different.
(c) The exact direction of the field can be predicted only at point C.
(d) The exact direction of the field can be predicted only at points A and C.
(e) The exact direction of the field cannot be predicted at any of the three
points.
Problem 3. Consider a horizontal square sheet, length L on each side, on which positive
charge is uniformly distributed with charge per unit area (surface charge density
21
FIG. 1: Charge distribution for Problem 2.
σ C/m2 ) (see Fig. 2). You measure the electric field at two points, each at a
height h = L/2 above the sheet: point C is directly above the center of the sheet
and point B is off center.
FIG. 2: A horizontal square sheet of charge for Problem 3.
Which one of the following statements is true about the field due to the finite
sheet of charge, observed at points B and C?
(a) The fields at points B and C have the same magnitude and same direction.
(b) The fields at points B and C have different magnitudes and different directions.
(c) The fields at points B and C have the same magnitude but different directions.
22
(d) The fields at points B and C have different magnitudes but the same direction.
(e) We cannot compare the fields at points B and C without knowing the numerical value of σ.
Problem 4. You perform two experiments (E1 and E2) in which you distribute charge +3Q
differently on an equilateral triangle made with thin insulating rods. E1: You
put identical charges, +Q each, in three localized blobs on the vertices of the
triangle. E2: You distribute charge +3Q uniformly on the triangle. The dashed
triangle in Fig. 3 shows an imaginary equilateral triangle concentric with the
insulating triangle. Which one of the following statements is true about the
electric field magnitude at points on the dashed imaginary triangle due to the
+3Q charge?
FIG. 3: Charge distribution for Problem 4.
(a) It is the same everywhere on the dashed triangle only in experiment E1.
(b) It is the same everywhere on the dashed triangle only in experiment E2.
(c) In each experiment, it is the same everywhere on the dashed triangle, but
the magnitudes differ in the two experiments.
(d) In each experiment, it is the same everywhere on the dashed triangle, and
the magnitudes are equal in the two experiments.
(e) Both in experiment E1 and in experiment E2 it varies from point to point
on the dashed triangle.
Problem 5. For Gauss’s law to be valid, the Gaussian surface used must be a
(a) highly symmetrical surface.
(b) spherical surface.
23
(c) cylindrical surface.
(d) open surface.
(e) closed surface.
Problem 6. Choose all of the following statements that must be true about a Gaussian
surface in order for Gauss’s law to be convenient for calculating the electric field
at a point on the surface:
(i) The electric field direction should be easy to predict at every point
on the surface.
(ii) The Gaussian surface must be chosen to take advantage of the symmetry of the charge distribution.
(iii) The Gaussian surface must be chosen to take advantage of the symmetry of the object enclosed inside it regardless of how the charges are
distributed on that object.
(a) (i) only.
(b) (ii) only.
(c) (iii) only.
(d) (i) and (ii) only.
(e) (i) and (iii) only.
Problem 7. In Fig. 4, a point charge +Q1 is at the center of an imaginary spherical surface
and another point charge +Q2 is outside it. Point P is on the surface of the
~ P be the electric
sphere. Let ΦS be the net electric flux through the sphere and E
field at point P on the sphere. Which one of the following statements is true?
FIG. 4: Charge distribution for Problem 7.
24
(a) Both charges +Q1 and +Q2 make nonzero contributions to ΦS but only the
~P .
charge +Q1 makes a nonzero contribution to E
(b) Both charges +Q1 and +Q2 make nonzero contributions to ΦS but only the
~P .
charge +Q2 makes a nonzero contribution to E
(c) Only the charge +Q1 makes a nonzero contribution to ΦS but both charges
~P .
+Q1 and +Q2 make nonzero contributions to E
~P .
(d) Charge +Q1 makes no contribution to ΦS or E
~P .
(e) Charge +Q2 makes no contribution to ΦS or E
Problem 8. Your friend measures the electric flux through three closed surfaces (1), (2), and
(3) to be 1 Nm2 /C, 2 Nm2 /C, and −3 Nm2 /C respectively. Choose all of the
following statements that can be inferred from these measurements:
(i) The area of surface (3) is largest.
(ii) The magnitude of the net charge enclosed inside surface (3) is largest.
(iii) The electric field everywhere on surface (1) is weaker than on surface
(2).
(a) (i) only.
(b) (ii) only.
(c) (i) and (ii) only.
(d) (ii) and (iii) only.
(e) (i), (ii), and (iii).
Problem 9. Shown in Fig. 5 are three concentric spherical Gaussian surfaces A, B, and C
with a positive point charge +Q at their center. A second, but negative point
charge −Q is enclosed only by surface C.
FIG. 5: Gaussian surfaces for Problem 9.
25
Which is a correct statement about the magnitudes of the electric flux ΦS
through the three surfaces?
(a) ΦA = ΦB = ΦC .
(b) ΦA = ΦB > ΦC .
(c) ΦA > ΦB > ΦC .
(d) ΦB > ΦA > ΦC .
(e) None of the above.
Problem 10. Choose all of the following cases for which the electric field at any point outside
the object can be calculated easily from Gauss’s law. (In each case, assume that
the insulators are nonpolarizable and no other charges are present anywhere.)
(i) Insulating sphere with a uniform charge throughout its volume.
(ii) Insulating dumbbell with a uniform charge throughout its volume.
(iii) Insulating dumbbell with only one of the two spherical balls at the
end uniformly charged throughout its volume.
FIG. 6: Diagram for Problem 10.
(a) (i) only.
(b) (ii) only.
(c) (i) and (ii) only.
(d) (i) and (iii) only.
(e) (i), (ii), and (iii).
Problem 11. Consider three possible Gaussian surfaces (a sphere, a cube, and a cylinder)
which extend half above and half below an infinite horizontal sheet of uniform
charge density as shown below in Fig. 7.
26
FIG. 7: Diagram for Problem 11.
Point A is located at the top center of each Gaussian surface. For which of the
Gaussian surfaces will Gauss’s law help us to easily calculate the electric field
at point A due to the sheet of charge?
(a) Only the sphere is symmetric enough.
(b) Only the cylinder, because the side walls have zero flux and it has circular
symmetry.
(c) Only the cylinder and the cube, because any shape with the side walls
perpendicular to the sheet and end caps parallel to the sheet will work.
(d) Only the sphere and the cylinder, because they have circular cross section.
(e) All surfaces will work since they are symmetric.
Problem 12. A thin insulating rod of length 1 m, with charge Q = +100 nC (nanocoulombs)
uniformly distributed over it, is symmetrically situated inside a spherical Gaussian surface with a total surface area of A = 15 m2 (see Fig. 8). No other charges
are present anywhere.
FIG. 8: Diagram for Problem 12.
We can use Gauss’s law to conclude that:
(i) the magnitude of the net electric flux through the Gaussian surface
is ΦS = Q/0 .
27
~ =
(ii) the electric field magnitude at any point on the surface is |E|
ΦS /A = Q/(0 A).
Which of these statements is true?
(a) (i) only.
(b) (ii) only.
(c) both (i) and (ii).
(d) neither (i) nor (ii).
(e) Not enough information.
Problem 13. The surface of a thin-walled cubic insulating (non-conducting) box is given a
uniformly distributed positive surface charge. Which one of the following can
be inferred about the electric field everywhere inside the insulating box due to
this surface charge using Gauss’s law?
(a) Its magnitude everywhere inside must be zero.
(b) Its magnitude everywhere inside must be nonzero but uniform (the same).
(c) Its direction everywhere inside must be radially outward from the center of
the box.
(d) Its direction everywhere inside must be perpendicular to one of the sides.
(e) None of the above.
Setup for Problems 14 and 15
You perform two experiments (see Fig. 9) in which you distribute charge +6Q differently
on the surface of an isolated hollow insulating (non-conducting) sphere: Experiment 1: You
put identical charges, +Q each, on the spherical surface in six localized blobs (you can
consider them point charges) such that the adjacent blobs are equidistant from each other.
Experiment 2: You distribute charge +6Q uniformly on the surface of the sphere.
Problem 14. In experiments 1 and 2, points A, B, and C are equidistant from the center and
lie in the same equatorial plane of the sphere. In experiment 1, points A and C
are straight out from two of the charges and point B is in between points A and
C as shown. Which one of the following statements is true about the electric
field magnitudes at points A, B, and C due to the +6Q surface charge?
28
FIG. 9: Diagram for Problems 14 and 15.
(a) In each experiment, the field magnitude is the same at points A, B, and C,
but the magnitudes differ in the two experiments.
(b) In each experiment, the field magnitude is the same at points A, B, and C,
and the magnitudes are equal in the two experiments.
(c) In experiment 1, the field magnitude is the same at points A, B, and C, but
not in experiment 2.
(d) In experiment 2, the field magnitude is the same at points A, B, and C, but
not in experiment 1.
(e) None of the above.
Problem 15. Which one of the following is a true statement about the electric field magnitude
inside the hollow insulating sphere due to the +6Q surface charge (see Fig. 9)?
(a) It is zero everywhere inside the sphere in both experiments.
(b) It is nonzero everywhere inside the sphere in both experiments.
(c) In experiment 1, it has a magnitude that varies from point to point inside
the sphere, but it is zero everywhere inside the sphere in experiment 2.
(d) In experiment 1, it has the same nonzero magnitude everywhere inside the
sphere, but it is zero everywhere inside the sphere in experiment 2.
(e) None of the above.
Problem 16. Six positive point charges, +Q each, are placed on an isolated hollow insulating
sphere such that the adjacent point charges are equidistant (same arrangement
as in experiment 1 in Problem 15). A spherical and a cubic Gaussian surface
concentric with the insulating sphere are shown in Fig. 10:
Choose all of the following statements that are true about the electric field due
to this charge distribution:
29
FIG. 10: Diagram for Problem 16.
(i) The electric field magnitude is the same everywhere on the cubic
Gaussian surface because the cube has the same symmetry as that of the
charge distribution.
(ii) The electric field magnitude is the same everywhere on the spherical Gaussian surface because the sphere has the same symmetry as the
insulating sphere.
(iii) The electric field is radially outward (straight out from the center)
everywhere on the spherical Gaussian surface.
(a) (i) only.
(b) (ii) only.
(c) (iii) only.
(d) (ii) and (iii) only.
(e) None of the above.
Problem 17. Shown below in Fig. 11 are three thin-walled insulating objects with a net charge
+Q uniformly distributed on their surfaces: a cube, a sphere, and an open ended
cylinder of length L (no caps) and diameter L. The objects are distant from
each other so that each may be considered electrically isolated.
We can easily use Gauss’s law to find the electric field due to the uniform surface
charge at a point outside due to:
(a) the cube only,
(b) the sphere only,
30
FIG. 11: Diagram for Problem 17.
(c) the sphere and cylinder only,
(d) the sphere and cube only,
(e) all three objects,
Problem 18. Choose all of the following statements that are true (Note: This question does
not refer to a particular charge distribution so a statement is true only if there
are no exceptions):
(i) If the electric field at every point on a Gaussian surface is zero, the
net electric flux through the surface must be zero.
(ii) If there is no charge enclosed inside a Gaussian surface, the electric
field everywhere on the surface must be zero.
(iii) If the net electric flux through a Gaussian surface is zero, the electric
field everywhere on the surface must be zero.
(a) (i) only.
(b) (ii) only.
(c) (i) and (ii) only.
(d) (i) and (iii) only.
(e) (ii) and (iii) only.
Setup for Problems 19 and 20
A cubic Gaussian surface with 1 m on a side is oriented with two horizontal and four vertical
faces, as shown in Fig. 12. It is in a uniform electric field of 20 N/C which is directed
vertically upward. Point A is on the top surface and point B on a side surface of the cubic
Gaussian surface.
31
FIG. 12: Diagram for Problems 19 and 20.
Problem 19. Which one of the following statements is true about the electric field at points
A and B?
(a) The field is zero at both points A and B.
(b) The field is zero at point A but not at point B.
(c) The field is zero at point B but not at point A.
(d) The field is nonzero at both points A and B and its direction is the same at
the two points.
(e) The field is nonzero at both points A and B but its direction is different at
the two points.
Problem 20. Choose all of the following statements that are true about the electric flux.
(i) The net flux through the whole cubic surface is zero.
(ii) The magnitude of the flux through the top face of the cubic surface
is 20 N m2 /C.
(iii) The magnitude of the net flux through the whole cubic surface is
20 N m2 /C.
(a) (i) only.
(b) (ii) only.
(c) (iii) only.
(d) (i) and (ii) only.
(e) (ii) and (iii) only.
32
Setup for Problems 21 and 22
Shown below in Fig. 13 are four imaginary surfaces coaxial with an isolated infinitely long
line of charge (with uniform linear charge density λ C/m):
(i) a closed cylinder of length L.
(ii) a sphere of diameter L.
(iii) a closed cubic box with side L.
(iv) a two dimensional square sheet with side L. The plane of the sheet is perpendicular to the line of charge.
FIG. 13: Diagram for Problems 21 and 22.
Problem 21. Choose all of the above surfaces through which the net electric flux is ΦS =
λL/0 :
(a) (i) only.
(b) (i) and (ii) only.
(c) (i) and (iii) only.
(d) (i), (ii), and (iii) only.
(e) (i), (ii), (iii), and (iv).
Problem 22. Choose all of the above surfaces which can be used as Gaussian surfaces to easily
find the electric field magnitude (due to the infinite line of charge) at a point P
shown on the surface using Gauss’s law:
(a) (i) only.
(b) (i) and (ii) only.
(c) (i) and (iii) only.
(d) (i), (ii), and (iii) only.
(e) (i), (ii), (iii), and (iv).
33
Problem 23. The diagram in Fig. 14 below shows the electric field lines in a region. Sadly, you
do not know the field inside the three regions i, ii, and iii. This cross-sectional
drawing is qualitatively correct.
FIG. 14: Diagram for Problem 23.
Which region (or regions) carries net charge of the greatest magnitude?
(a) (i) only.
(b) (ii) only.
(c) (iii) only.
(d) (ii) and (iii) which have equal net charge.
(e) (i), (ii), and (iii) which have equal net charge.
Problem 24. Shown below in Fig. 15 are four regions A, B, C, and D (separated by spherical
surfaces). The electric field is zero in regions A (innermost) and D (outermost).
The electric field in regions B and C is radially outward and inward, respectively.
Choose all of the following statements that must be true:
(i) The combined net charge enclosed in all the regions shown must be
zero.
(ii) There cannot be a point charge at the center of region A.
(iii) There must be a negative surface charge between regions B and C.
(a) (i) only.
(b) (i) and (ii) only.
(c) (i) and (iii) only.
34
FIG. 15: Diagram for Problem 24.
(d) (ii) and (iii) only.
(e) (i), (ii), and (iii).
Problem 25. In Fig. 16 below, a point charge +Q is near a thin hollow insulating (nonconducting) sphere of radius L that has an equal amount of charge +Q uniformly
distributed on its surface. No other charges are around.
FIG. 16: Diagram for Problem 25.
Which one of the following is a true statement about the net electric field (due
to the point charge and the surface charge on the hollow insulating sphere) at
points A (outside the sphere at a distance 2L from the center) and B (inside
the sphere at a distance L/2 from the center)?
(a) The electric field is zero at point A but is nonzero at point B.
(b) The electric field is nonzero at point A but is zero at point B.
(c) The electric field is nonzero at both points A and B.
35
(d) The electric field is zero at both points A and B.
(e) It is impossible to answer this question without knowing the numerical value
of Q.
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